Logarithmic Functions
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Questions and Answers

What is the domain of the function $f(x) = \log_2 x$?

  • Negative real numbers
  • All real numbers
  • Non-real numbers
  • Positive real numbers (correct)
  • What is the range of the function $f(x) = \log_5 x$?

  • Non-real numbers
  • Positive real numbers
  • All real numbers (correct)
  • Negative real numbers
  • What are the x-intercepts of the functions $f(x) = \log_3 x$ and $g(x) = \log_{10} x$?

  • (1, 0) and (1, 0) (correct)
  • (0, 1) and (0, 1)
  • (0, 1) and (1, 0)
  • None of the above
  • What is the y-intercept of the function $f(x) = \log_7 x$?

    <p>There is no y-intercept</p> Signup and view all the answers

    What is the vertical asymptote for the function $f(x) = \log_4 x$?

    <p>x = 0</p> Signup and view all the answers

    What is the domain of the function $f(x) = rac{1}{2} imes ext{log}_2 x$?

    <p>All positive real numbers except 0</p> Signup and view all the answers

    Which of the following is true about the function $f(x) = ext{log}_5 x$?

    <p>It is a decreasing function</p> Signup and view all the answers

    What is the y-intercept of the function $f(x) = 1.5 imes ext{log}_3 x$?

    <p>There is no y-intercept</p> Signup and view all the answers

    What is the vertical asymptote for the function $f(x) = ext{log}_{10} x$?

    <p>There is no vertical asymptote</p> Signup and view all the answers

    Which of the following is true about the function $f(x) = ext{log}_2 x$?

    <p>It has a range of all real numbers</p> Signup and view all the answers

    Study Notes

    Logarithm Functions Overview

    • The domain of ( f(x) = \log_2 x ) is all positive real numbers, expressed as ( x > 0 ).
    • The range of ( f(x) = \log_5 x ) encompasses all real numbers, represented as ( (-\infty, \infty) ).

    x-intercepts of Logarithmic Functions

    • The x-intercept of ( f(x) = \log_3 x ) occurs at ( x = 1 ) since ( \log_3 1 = 0 ).
    • The x-intercept of ( g(x) = \log_{10} x ) is also at ( x = 1 ) for the same reason, ( \log_{10} 1 = 0 ).

    y-intercepts of Logarithmic Functions

    • The function ( f(x) = \log_7 x ) does not have a y-intercept since it only exists for ( x > 0 ).
    • For ( f(x) = 1.5 \cdot \log_3 x ), the y-intercept cannot be defined as logarithmic functions do not cross the y-axis.

    Vertical Asymptotes of Logarithmic Functions

    • The vertical asymptote for ( f(x) = \log_4 x ) is at ( x = 0 ).
    • The vertical asymptote for ( f(x) = \log_{10} x ) is also at ( x = 0 ).

    Properties of Logarithmic Functions

    • For ( f(x) = \log_5 x ), it is true that as ( x ) increases, ( f(x) ) increases without bound, and it approaches negative infinity as ( x ) approaches zero from the positive side.
    • For ( f(x) = \log_2 x ), it is generally true that the function is increasing and never crosses the x-axis except at ( x = 1 ).

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    Description

    Explore the characteristics of logarithmic functions and compare the graphs of log base 2, log base 3, log base 5, and log base 10. Understand the domain, range, x-intercept, and y-intercept of logarithmic functions.

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